The Gaussian approximation for multi-color generalized Friedman’s urn model

The generalized Friedman’s urn model is a popular urn model which is widely used in many disciplines.In particular,it is extensively used in treatment allocation schemes in clinical trials.In this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman’s urn model with both homogeneous and non-homogeneous generating matrices.The Gaussian process is a solution of ...     

The Gaussian approximation for generalized Friedman’s urn model with heterogeneous and unbalanced updating

The Friedman’s urn model is a popular urn model which is widely used in many disciplines.In particular,it is extensively used in treatment allocation schemes in clinical trials.Its asymptotic properties have been studied by many researchers.In literature,it is usually assumed that the expected number of balls added at each stage is a constant in despite of what type of balls are selected,that is,the updating of the urn is assumed to be balanced.When it is not,the asymptotic property of the Friedman’s urn model is stated in the book of Hu and Rosenberger(2006) as one of open problems in the area of adaptive designs.In this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a general multi-color Friedman type urn model with heterogeneous and unbalanced updating.The Gaussian process is a solution of a stochastic differential equation.As an application,we obtain the asymptotic properties including the asymptotic normality and the exact law of the iterated logarithm.     

Solving the 0–1 proportional knapsack problem by sampling

In this paper, we deal with the proportional knapsack problem that is a variation on the ordinary knapsack problem. In the proportional knapsack problem, we look at filling an urn with objects having two characteristics: color and weight. The colors of the objects in the urn should be proportional to the distribution of the colors in the object universe, and the total weight of the objects in the urn should be as close as possible to the capacity of the urn. The formulation of the problem was motivated by a real-life application from the area of finance, called a dollar roll. We show that the proportional knapsack problem is NP-hard, and then, using sampling, develop a heuristic procedure for solving the problem.Partial support from the Fund for the Promotion of Research and from the Alexander Goldberg Memorial Fund at Technion is gratefully acknowledged.     

Solution to the OK Corral Model via Decoupling of Friedman's Urn

We consider the OK Corral model formulated by Williams and McIlroy(11) and later studied by Kingman.⁽⁷⁾ In this paper we refine some of Kingman's results, by showing the connection between this model and Friedman's urn, and using Rubin's construction to decouple the urn. Also we obtain the exact expression for the probability of survival of exactly S gunmen given an initially fair configuration.     

Urn模型的B网及其应用

Urn模型的离散概率分布被用于构造逼近格式,具有Bernstein多项式的许多特性。本文用B网技术对Urn模型概率分布给出新的解释,指出了Urn模型B网的诸多性质,导出分布函数曲线的导数公式,构造了Urn分布的对偶基及与此相关的拟插值算子。     

Statistical modeling for discrete patterns in a sequence of exchangeable trials

This paper proposes a new method for constructing a sequence of infinitely exchangeable uniform random variables on the unit interval. For constructing the sequence, we utilize a Pólya urn partially. The resulting exchangeable sequence depends on the initial numbers of balls of the Pólya urn. We also derive the de Finetti measure for the exchangeable sequence. For an arbitrarily given one-dimensional distribution function, we generate sequences of exchangeable random variables with the one-dimensional marginal distribution by transforming the exchangeable uniform sequences with the inverse function of the distribution function. Among them we mainly investigate sequences of exchangeable discrete random variables. They differ from the well-known exchangeable sequence generated only by the Pólya urn scheme. Some examples are also given as applications of the results to exact distributions of some statistics based on sequences of exchangeable trials. Further, from the above exchangeable uniform sequence we construct partial or Markov exchangeable sequences. We also provide numerical examples of statistical inference based on the exchangeable and Markov exchangeable sequences.     

An analytic approach for the analysis of rotations in fringe-balanced binary search trees

This paper presents an analytic approach to the construction cost of fringe-balanced binary search trees. In [7], Mahmoud used a bottom-up approach and an urn model of Pólya. The present method is top-down and uses differential equations and Hwang's quasi-power theorem to derive the asymptotic normality of the number of rotations needed to construct such afringe balanced search tree. We also obtain the exact expectation and variance with this method. Although Pólya's urn model is no longer needed, we also present an elegant analysis of it based on an operator calculus as in [4].This research was supported by the Austrian Research Society (FWF) under the project number P12599-MAT.     

Properties of analytic Morrey spaces and applications

In this note, we aim to study analytic Morrey spaces . We first give the canonical factorization for . Then by applying p‐Carleson measure, we prove an atomic decomposition theorem of . As an application of the decomposition theorem, the interpolation problem of is solved. Finally, we show the boundedness and compactness of Toeplitz operators on .     

Anti‐periodic solutions for a gradient system with resonance via a variational approach

In this paper, we investigate a second‐order resonance anti‐periodic boundary value problem where is the m‐th eigenvalue of the corresponding eigenvalue problem. By using the dual least action principle, we obtain an existence result. In addition, we obtain the existence of 2T‐periodic solutions for .     

On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given by (1) In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞ ‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian (2) We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞ . We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.     

Solutions of the congruence

In this paper we characterize, in terms of the prime divisors of n, the pairs for which n divides . As an application, we derive some results on the sets for some choices of f.     

Almost resolvable k‐cycle systems with

In this paper, we show that if and , then there exists an almost resolvable k‐cycle system of order for all except possibly for and . Thus we give a partial solution to an open problem posed by Lindner, Meszka, and Rosa (J. Combin. Des., vol. 17, pp. 404–410, 2009).     

Central and Functional Central Limit Theorems for a Class of Urn Models

We consider an approach based on tails to certain central limit and functional central limit theorems for a class of two color urn models. In particular, some of the results are derived from an associated Ornstein–Uhlenbeck process, and for another result we give an alternative proof based on martingale tails.      

Gradient estimate for solutions of nonlinear singular elliptic equations below the duality exponent

In this paper, we study the homogeneous Dirichlet problem for an elliptic equation whose simplest model is where , N≥3 is an open bounded set, θ∈]0,1[, and f belongs to a suitable Morrey space. We will show that the Morrey property of the datum is transmitted to the gradient of a solution.     

On the bifurcation results for fractional Laplace equations

In this paper, we consider the bifurcation problem for the fractional Laplace equation where is an open bounded subset with smooth boundary,  stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue λ₁ of the problem and, conversely.     

A compactness criterion and application to the commutators associated with Schrödinger operators

Let T be an integral operator. In this paper, we introduce a ‐compactness criterion of , where . As an application, we apply this criterion to deal with ‐compactness of commutators associated to Schrödinger operators with potentials in the reverse Hölder's class.     

On Schauder equivalence relations

We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show that and are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotient where T is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality .     

Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetrie decreasing re arrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing ?. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.     

On semi‐isogenous mixed surfaces

Let C be a smooth projective curve and G be a finite subgroup of whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of . Let be the index two subgroup . If G⁰ acts freely, then is smooth and we call it semi‐isogenous mixed surface. In this paper we give an algorithm to determine semi‐isogenous mixed surfaces with given geometric genus, irregularity and self‐intersection of the canonical class. As an application we classify irregular semi‐isogenous mixed surfaces with and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with . We provide an example of a minimal surface of general type with and .     

A time-dependent version of Pólya's urn

A process is defined that consists of drawing balls from an urn and replacing them along with extra balls. The process generalizes the well-knownPólya urn process. It is shown that the proportion of red balls in the urn converges to a random limit that may have a nonzero probability of being 0 or 1, but is nonatomic elsewhere.